ME8230 Nonlinear Dynamics Lecture 1, part 1 Introduction, some basic math background, and some random examples Prof. Manoj Srinivasan Mechanical and Aerospace Engineering srinivasan.88@osu.edu Spring mass damper system LINEAR (unforced) Spring Mass mẍ + kx = 0 (undamped) Damper Spring Mass mẍ + cẋ + kx =0 (damped) ( You can add external forcing terms to all the unforced examples in this lecture )
Spring mass damper system NON-LINEAR Spring Mass undamped mẍ + f(x) =0 Damper Mass Spring (unforced) damped f(x) = Nonlinear spring force g(xdot) = Nonlinear damping more general nonlinear damped mass-spring system mẍ + g(ẋ)+f(x) =0 mẍ + h(x, ẋ) =0 Nonlinear spring-mass examples x(t) Spring Mass mẍ + f(x) =0 can be a nonlinear model of any of these systems on the right when the displacements are large enough... (but mostly dominated by the first mode) x(t) x(t) Displacements small enough implies Linear spring approximation would be good enough (usually).
Common Nonlinear Stiffness behaviors Softening spring Linear spring Force f(x) Hardening spring Hardening spring Linear spring Softening spring f(x) = kx+ax 3, a>0, k>0 f(x) = kx f(x) = kx-ax 3 Displacement x Different nonlinearities might lead to different namical behaviors Ex1. which of the above cubic f(x) correspond to hardening and softening springs? Ex2. Model mass hanging from a taut horizontal string. Nonlinear damping ẋ examples F = Drag Damping Friction Linear damping: F = cẋ (negative sign indicates drag opposes velocity) Fluid drag at large Re: Coulomb friction: dry friction F = cẋ ẋ = c(ẋ) 2 sign(ẋ) if ẋ = 0, µ s N apple F apple µ s N if ẋ 6= 0, F = µ d N sign(ẋ) Material damping at high enough strain rates is nonlinear
Many degrees of freedom systems LINEAR MẌ + CẊ + KX =0 K - stiffness matrix M - mass matrix C - damping matrix 2 x 1 x n Equations obtained from FEM, modal analysis, etc. E.g., trusses/beam/plate/other elastic structures undergoing small deformations 3 x 2 X = 6. 7 4. 5 NON-LINEAR MẌ + H(X, Ẋ) =0 Pretty general: most smooth nonlinear (discrete, unforced) mechanical systems have such equations Forced finite dimensional mechanical systems pretty general mechanical system special case e.g., double pendulum with joint torques
Other Examples Simple pendulum O ml 2 + mgl sin =0 gravity g length l + g l sin =0 A mass m
Aeroelastic oscillations Tacoma Narrows Bridge Collapse Nonlinear (negative) damping Hopf bifurcation Limit cycles NASA Tail Flutter Test Motions of disks and Equations of motion for a cylinder rolling without slip cylinders
Double pendulum See also MATLAB example (Hamiltonian) Chaos Sensitive dependence on initial conditions Lyapunov exponents Double pendulum with torque actuators at each joint and PID control Robot arm See MATLAB example
Robots and humans Hybrid (piecewise smooth) systems Non-smooth systems Limit cycles Stability, Linearized Collisions ODEs corresponding to mechanical systems tend to be (most naturally) 2nd order ODEs Highest derivative order = 2
Example High (e.g., 2nd) order ODEs to first order ODEs always possible to convert Say you are given 2nd order ODE: Introduce new variables for all derivatives of x, except the highest order (=2) x 1 = x x 2 =ẋ mẍ + kx =0 Write an ODE for each of the new variables ẋ 1 = x 2 ẋ 2 = kx 1 /m What is this course Mostly, ordinary differential equations of the form about? t = independent variable y = dependent variable (can be scalar or vector) = f(y, t) 2 y 1 y n 3 y 2 y =. 6. 7 4. 5
What to do with PDEs? Partial Differential Equations Simplify, Discretize, Project (e.g., finite elements, finite volume, finite differences, other Galerkin projections, lumped parameter approximations) Ordinary Differential Equations Some of the ideas in the course actually carry over to PDEs without reducing them to ODEs, but we will not discuss such. Linear ODEs = f(y, t) Linear time-varying inhomogeneous (non-homogeneous) variable coefficients = A(t)y + B(t) Linear time-varying homogeneous (variable coefficients) = A(t)y Linear time-invariant homogeneous (constant coefficients) = Ay Inhomogeneous means B(t) is not zero Homogeneous means B(t)=0 A is a constant and not time-varying
Linear superposition for linear homogeneous ODE = A(t)y If y 1 (t) and y 2 (t) are solutions, then y 3 = y 1 (t)+y 2 (t) is also a solution. 1 (t) 2 (t) = A(t)y 1 (t) = A(t)y 2 (t) add Thus y 3(t) d(y 1 (t)+y 2 (t)) = A(t)y 3 (t) = A(t)y 1 (t)+a(t)y 2 (t) = A(t)(y 1 (t)+y 2 (t)) Linear superposition for linear inhomogeneous ODE = A(t)y(t)+B(t) B(t) - Input (the inhomogeneous term ) y(t) - Output Say y 1 (t) is a solution when B 1 (t) istheinput. Say y 2 (t) is a solution when B 2 (t) istheinput. y = 1 y 1 + 2 y 2 is a solution when B = 1 B 1 + 2 B 2. ẏ 1 = A(t)y 1 + B 1 (t) ẏ 2 = A(t)y 2 + B 2 (t) add ẏ 1 +ẏ 2 = A(t)(y 1 + y 2 )+(B 1 (t)+b 2 (t))
Key properties of linear systems Superposition of solutions proportionality (input and output scale together). Various analytical and semi-analytical techniques available for periodic, non-periodic forcing. In the absence of forcing, closed-form solutions known. e.g., Laplace transforms, Fourier series, Green s functions, modal analysis, etc. Nonlinear systems have none of these nice properties, in general. This course is about making sense of nonlinear ODEs by other means.